PEOFESSOE CAYLEY ON PEEPOTENTIALS. 
761 
iv 2 . . .+£ 2= =7’ 2 , div . . . dt=r s ~ s '~ 1 dr dS, where dS is the element of surface of the (s-s 1 )- 
coordinal unit-sphere £ 2 . . . -j-£ 2 =l. We thus obtain 
V= 
p r s-s'~ lfo 
){A 2 + r 2 p s+s 
where the integral in regard to r is taken from 0 to go , and the integral J dS over the 
surface of the unit-sphere ; hence by Annex I. the value of this last factor is = \ 2j — f . 
The integral represented by the first factor will be finite, provided only \s’ -\-q be positive; 
which is the case for any value whatever of s' if only q be positive. 
The first factor is an integral such as is considered in Annex II. ; to find its value we 
have only to write r = A x, and we thus find it to be 
1 n T” xi s -^ s '~ l dx . 1 ^r^(s— + g ) 
— ( A 2)^+? 2 Jo (X V1Z,= A s ' +2? ‘ r(is + gr) ’ 
and we thus have 
v L (ra-T&'+g) 
v- a , +29 . T{¥+q) » 
(rp-Tay+g) i # 
r(i* + g) \ {a -xf . . . + [c-zf + id p 5 '* 9 
137. As a verification observe that the prepotential equation □ V=0, that is 
/ d z ,^1 , i d 2 , d* ,2q + l <Ay__n. 
\dcd * dc* dd 2 ' de 2 du 2 u du) ’ 
for a function V which contains only the s'+l variables (a . . .c, u) becomes 
. \d 2 ,d 2 . 2q+\ d\y_Q 
\dcd ‘ dc z . dud u du) 5 
which is satisfied by V a constant multiple of \{a—x) 2 . . J r {c—zf-\-id\^~ s '~ q . 
Annex IX. The Gauss-Jacobi Theory of Episplieric Integrals. — No. 138. 
138. The formula obtained (Annex IV. No. 110) is proved only for positive values 
of m ; but writing therein ^=0, m = — ^ , it becomes 
dx ... dz 
57 
T dt.tr 1 ( 
1 t . , 
_ C 2 e\ 
J \ 
< t+f 
t + fd t) 
a formula which is obtainable as a particular case of a more general one 
dS 
(- ^ - 2 ( r wr di 1 
) \{*Jx...z, w) 2 f r(i«)J_ A — Disct. { (*XN • • • Z, W, T 
r) 2 +^(X 2 ...+z 2 +w 2 +T°-)^ 
